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Rewrite the function by completing the square. {:[f(x)=x^(2)-14 x+63],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−14x+63 f(x) = x^2 - 14x + 63 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−14x+63 f(x) = x^2 - 14x + 63 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function:\newlinef(x) = x2−14x+63x^2 - 14x + 63\newlineTo complete the square, we need to form a perfect square trinomial from the quadratic and linear terms.
  2. Completing the square: First, we identify the coefficient of the xx term, which is −14-14. We then take half of this coefficient and square it to find the number that we need to add and subtract to complete the square.(−142)2=(−7)2=49\left(\frac{-14}{2}\right)^2 = (-7)^2 = 49
  3. Identifying the coefficient: We add and subtract 4949 inside the function to complete the square, making sure the function's value doesn't change.\newlinef(x)=(x2−14x+49)+63−49f(x) = (x^2 - 14x + 49) + 63 - 49
  4. Adding and subtracting to complete the square: Now we factor the perfect square trinomial and simplify the constants.\newlinef(x) = (x−7)2+14(x - 7)^2 + 14
  5. Factoring the perfect square trinomial: We have successfully rewritten the function by completing the square.\newlineThe completed square form of the function is f(x)=(x−7)2+14f(x) = (x - 7)^2 + 14.

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